Chaos - James Gleick [124]
Asked to organize an attack on such a system, a traditional physicist might begin by making as complete a physical model as possible. The processes governing the creation and breaking off of drips are understandable, if not quite so simple as they might seem. One important variable is the rate of flow. (This had to be slow compared to most hydrodynamic systems. Shaw usually looked at drop rates of 1 to 10 per second, which meant a flow rate of 30 to 300 gpf—gallons per fortnight.) Other variables include the viscosity of the fluid and the surface tension. A drop of water hanging from a faucet, waiting to break off, assumes a complicated three-dimensional shape, and the calculation of this shape alone was, as Shaw said, “a state-of–the-art computer calculation.” Furthermore, the shape is far from static. A drop filling with water is like a little elastic bag of surface tension, oscillating this way and that, gaining mass and stretching its walls until it passes a critical point and snaps off. A physicist trying to model the drip problem completely—writing down sets of coupled nonlinear partial differential equations with appropriate boundary conditions and then trying to solve them—would find himself lost in a deep, deep thicket.
An alternative approach would be to forget about the physics and look only at the data, as though it were coming out of a black box. Given a list of numbers representing intervals between drips, could an expert in chaotic dynamics find something useful to say? Indeed, as it turned out, methods could be devised for organizing such data and working backward into the physics, and these methods became critical to the applicability of chaos to real-world problems.
But Shaw began halfway between these extremes, by making a sort of caricature of a complete physical model. Ignoring drop shapes, ignoring complex motions in three dimensions, he roughly summarized drip physics. He imagined a weight hanging from a spring. He imagined that the weight grew steadily with time. As it grew, the spring would stretch and the weight would hang lower and lower. When it reached a certain point, a portion of the weight would break off. The amount that would detach, Shaw supposed arbitrarily, would depend strictly on the speed of the descending weight when it reached the cutoff point.
Then, of course, the remaining weight would bounce back up, as springs do, with oscillations that graduate students learn to model using standard equations. The interesting feature of the model—the only interesting feature, and the nonlinear twist that made chaotic behavior possible—was that the next drip depended on how the springiness interacted with the steadily increasing weight. A down bounce might help the weight reach the cutoff point that much sooner, or an up bounce might delay the process slightly. With a real faucet, drops are not all the same size. The size depends both on the velocity of the flow and on the direction of the bounce. If a drop starts off its life already moving downward, then it will break off sooner. If it happens to be on the rebound, it will be able to fill with a bit more water before it snaps. Shaw’s model was exactly crude enough to be summed up in three differential equations, the minimum necessary for chaos, as Poincaré and Lorenz had shown. But would it generate as much complexity as a real faucet? And would the complexity be of the same kind?
Thus Shaw found himself sitting in a laboratory in the physics building, a big plastic tub of water over his head, a tube running down to a premium-quality hardware-store brass nozzle. As each drop fell, it interrupted a light beam, and a microcomputer in the next room